Weak cop number from excluded asymptotic minors

Establish whether every locally finite connected graph that excludes a finite planar graph as an asymptotic minor has finite weak cop number, and, if so, determine whether the weak cop number is bounded by a function of the maximum treewidth of finite asymptotic minors, possibly with the identity function as such a bound.

Background

Theorem 3.1 gives a lower bound on the weak cop number in terms of the treewidth of finite asymptotic minors. The question asks for a converse-type upper bound under exclusion of a finite planar asymptotic minor. The authors note that local finiteness is necessary, because they provide a non-locally-finite counterexample with treewidth 2 and infinite weak cop number.

References

Is it true that if a locally finite connected graph $G$ excludes some finite planar graph $H$ as an asymptotic minor, then it has finite weak cop number? If so, does there exist some function $f:\mathbb N\to \mathbb N$ such that $\wco(G)\leq f\big(\max\sg{\tw(H): |H|<\infty~\text{and}~H\preceq_{\infty} G}\big)?$ Can we choose $f=\mathrm{id}_{\mathbb N}$?

— Coarse cops and robber in graphs and groups  (2502.15571 - Esperet et al., 21 Feb 2025) in Question (q: minors), Section 3.1, subsection “Asymptotic minors and weak cop number”